Quasiconvex functions, <span class="mathjax-formula">$SO(n)$</span> and two elastic wells

A NNALES DE L ’I. H. P., SECTION C

K ^EWEI Z ^HANG

Quasiconvex functions, SO(n) and two elastic wells

Annales de l’I. H. P., section C, tome 14, n

^o

6 (1997), p. 759-785

<http://www.numdam.org/item?id=AIHPC_1997__14_6_759_0>

© Gauthier-Villars, 1997, tous droits réservés.

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Quasiconvex functions, SO(n) ^{and two elastic wells}

Kewei ZHANG Ann. Inst Henri Poincaré,

Vol. 14, ^n° 6, 1997, p. 759-785 Analyse ^{non linéaire}

School of Mathematics, Physics, Computing ^and electronics, Macquarie University, ^NSW 2109, ^Australia.

E-mail: [email protected]

ABSTRACT. - We use

approximations

^of

minimizing

sequences to

study

^the

growth

^{of some}

quasiconvex

^{functions near their zero sets.}

We show that for

SO(n) ,

^the

quasiconvexification

^{of the distance function}

SO (n) )

^can be bounded below

by

the distance function itself. In certain cases of the

incompatible

^two elastic well structure, ^we establish

a similar result. We also prove that for small

Lipschitz perturbations

^of

SO(n)

and of the two well structure, the

Young

^{measure limits of}

gradients supported

^{on these}

perturbed

^{sets are Dirac masses.}

1. INTRODUCTION

The

study

^of

Young

^{measures limit of}

gradients [KP, BFJK] supported

in various

compact

^{sets in}

MN n and quasiconvex

relaxations of certain distance functions to these sets are very

important subjects

^{in the}

study

^of

martensitic

phase

transitions and

optimal design problems (see [CK, BJl, BJ2, F, K, KS]).

^{As far as I}

know, explicit

relaxation formulas are hard to obtain and there are

only

^{a few known}

examples [Dal, Da2, KS, K, D2, DR]. Therefore,

^the

study

of the behaviour of

quasiconvex

functions is very difficult because we cannot work on them

directly.

^{It is}

closely

^{related to}

the

analysis

^of

quasiconvex

hulls of the

neighbourhoods

^{of the zero sets for}

given non-negative

functions and the

stability problems

^{related to}

Young

Annales de l’Institut Henri Poincaré - Analyse ^non lineaire - 0294-1449 Vol. 14/97/06/@ Elsevier, ^Paris

760 K. ZHANG

measures under

perturbations.

^In this paper, ^we

give

^some

examples

^where

we can estimate the

growth

of certain

quasiconvex

functions of the form

for ^some closed subset K C where is the space of all N x n real

matrices,

^without

knowing

^{the exact} formula of

F(P),

^and

where

Qdistp (P, K)

^{is the}

quasiconvexification

^of

^distp (P, K),

^{and p > l.}

We establish these results for

SO(n)

^{and for two}

incompatible

^elastic

wells

SO(n)

^U

SO(n)H

when H satisfies the technical condition

(1.3)

below. The

p-th

power of ^a distance function to a

given compact

^{set is} the

simplest

function from

geometric point

of view and the

quasiconvex

relaxations of this

type

of functions

give

information of

quasiconvex

^hulls

of the

neighbourhoods

^{of the zero sets of the}

original

^functions.

The main results are the

following.

THEOREM 1.1. -

Suppose n 2:

^{2. Let}

F(P)

⁼

Qdist2(P, SO(n))

be the

quasiconvexification (cf. [Da 1, Da2],

^{also see}

Definition ^{2.1 )} of

dist2(P, SO(n)),

^{P E Mnxn.} Then there exists a constant

c(n)

^>

^0,

^such

that

for all

^{P E}

The method we use for

proving

of Theorem 1.1

applies

^{to two}

incompatible

elastic wells under a further technical condition. Let n ^> 3 and

where

SO(n)H

⁼

{RH, R E SO (n) ~,

^{and H is a n x n}

positive

^definite

diagonal

^matrix

satisfying

It was

proved

ⁱⁿ

[Ma, Sv]

^{that the}

quasiconvex

^{hull of K defined}

by (1.2)

remains K itself under

assumption (1.3).

^We

have,

COROLLARY 1.2. -

Suppose n 2 3,

^{K is}

given by (1.2)

^{with H}

satisfying (1.3). Then,

there exists ^a

positive

^constant

c(n, H)

^> 0 such that

We also

study

^the

Young

^{measure limit of}

gradients ([KP, BFJK]) supported

^’near’

SO(n)

^and

SO(n)

^U

SO(n)H.

^We

give

^{some estimates}

linéaire

QUASICONVEX FUNCTIONS, SO(n) ^AND TWO ELASTIC WELLS 761

of the

quasiconvex

hull of the

E-neighbourhood

^{of these sets. This}

problem

is related to the

stability

of ’one-well’ and two

incompatible

^{well structure}

when n ⁼ 3. For a

compact

^{set K C}

MNxn,

^let

and let be the

quasiconvex

^{hull of}

^Ka

^{for a > 0. We have}

COROLLARY 1.3. - Assume

(1.3).

There exist constants

C(n)

^>

^0, C(n, H)

^>

^0,

^{such that}

and

for

^{all E > 0.}

We

always

^{assume that H c IRn is a} bounded open and connected ^set with smooth

boundary.

^Let

f : SO(n) -~

^Mn"n

and g : SO(n)

^U

be

Lipschitz

functions such that

for some E > 0.

We have

THEOREM 1.4. -

Suppose

^that

f

^and g ^are

defined

^as

^above,

^let

K f

^and

Kg

^{be their}

graphs, respectively,

^i.e.

Let be a bounded sequence in 1 _{p ~,} such that u~ ^{~ u in and}

dist(Duj, I~ f ) -~

⁰

(dist(Duj, K9 ) -~ ^0,

respectively)

^almost

everywhere as j

^{~ ~.}

^Then, for sufficiently

^small

E > 0 and _up to a

subsequence,

^{Du almost}

everywhere.

^{In other}

words, Kf

^and

Ig

^support

only

^trivial

Young

^{measure limit}

of gradients (see

Theorem 2.4

for

^the

definition of Young measures).

Similar to Theorem

l.l,

^we have

COROLLARY 1.5. -

Suppose

^{that n >}

^2,

^{and let}

f

^and g

satisfy ( 1.5)

^and

(1.6), respectively, for

^{E > 0}

sufficiently

small. Let

F(P)

⁼

Qdist2(P, Kf),

Vol. 14, nO 6-1997.

762 K.ZHANG

and let

G(P)

⁼

Qdist2(P,kg)

^{be the}

quasiconvexifications of dist2 (P, K f)

and

dist2(P, respectively.

Then there exists a constant

c(n, E)

^{> 0}

such that

for

^{all P E}

The methods we use to prove Theorem 1.1 ^are:

(i)

^an

approximation

^of

W 1 ~ 2 minimizing

sequences

by ^{YT~ l ~‘~}

sequences

(Lemma 3.1), using

^the

maximal function method

through

^a modified version of

[Z,

^Lemma

3.1];

(ii)

^V.

Sverak’s

idea of

showing

^{that a} sequence is ^a

Cauchy

sequence

[Sv].

To prove Theorem 1.1 and

Corollary ^1.2,

^{we have to make use of the}

special

structure of the sets

concerned,

^that

is,

^both

SO(n)

^and

SO(n)

^U

SO(n)H

have the

property

for some a >

0,

^where

adjP

^{is the}

transpose

of the cofactors of P E Mnxn

(see [Sv]).

^{In order to use this}

property,

the sequences under

study

^{should be}

in at least. When n ^>

3,

^our sequence is bounded in

therefore,

we have to

approximate

^the

original

sequence

by

^a bounded sequence in

Wl°°.

I do not known how to

bypass

^Lemma 3.1 and prove the result

directly.

^The

approximation

^lemma

(Lemma 3.1)

^{stands on its own}

right.

It

gives

^an estimate of the minimum energy of

quasiconvex

relaxations for distance functions

distp ( ~, K)

^for

general compact

^{sets K c}

M N x n

when

we

only

minimize the energy ^{on a set} of bounded functions.

In

§2,

notation and

preliminaries

^are

given

^which will be used to prove ^our main results.

§3

is devoted to

establishing

^the

approximation

lemma which is crucial to the

proof

of Theorem 1.1. In

§4,

^we prove the results stated above.

To conclude this

section,

^we

justify

that the function

dist2 (P, SO (n) )

^is

not

quasiconvex

itself. In

fact,

^{it is not trivial to} prove it.

PROPOSITION 1.6. -

,S’O (n) ) : quasiconvex.

Proof. -

^Let

f (P) = min~~P - A12, ~P -

^where

A,

^{B E}

are fixed matrices. It was established in

[K]

^{that the}

quasiconvexification

Q f

^of

f

^{has the}

explicit

^form

QUASICONVEX FUNCTIONS, WO(n) ^{AND TWO} ELASTIC WELLS 763

where is the

greatest eigenvalue

of the matrix

(A - B)T (A - ^B).

Now, set .4 ⁼

7,

B ⁼ .J, where I is the n x ii,

identity ^matrix,

where 12

^and

I" _~~

^{are 2 x 2 and}

(7z - 2)

^x

(n - 2) identity

^matrices

respectively.

^Consider

F(P) = I ~2, ~P -

^{P E}

^Mnxn,

^we

have

F(P) > ,5’C)(71,)),

^since

~,

^{J E}

SO(n).

^If

SO(n))

was

quasiconvex, by

Definition 2.2

below, QF(P) > dist2(P, SO(n)),

^for

every P ^E 1’tI " X " . In

particular, QF(0) > dist2(0, SO(n) ),

where 0 is the

n x zero matrix. We have

and if ^we notice that ⁼ 4 when A ⁼

I,

^{B =}

J,

^{we have}

Contradiction. The

proof

^is

complete.

⁰

2. NOTATION AND PRELIMINARIES

Throughout

^{the rest of this}

.paper 0

^{is a} bounded open subset of IRn. We denote

by

the space of real N ^{x n} matrices with the

IRNn metric;

hence the ^norm of P E is defined

by I

^P

~_

^where

tr is the trace

operator

^and

PT

is the

transpose

of P. The inner

product

of two matrices in

MNxn

is P.

Q

⁼

trPTQ.

^{For an n x n matrix}

^P,

denote

by adjP

^the

transpose

of the cofactors of P.

SO ( n)

^{is the set of}

all rotations with determinant 1. For ^a

compact

^{subset K C}

MNxn,

^let

convK,

^{diamK be the convex}

hull,

diameter and the ^norm of

K, respectively,

^where

We write for the space of continuous functions

(~ : ~ 2014~

^!R

having compact support

ⁱⁿ

0,

and define ⁼

C 1 ( SZ )

^{n If}

Vol. 14, n° 6-1997.

764 K. ZHANG

1 p

^{cxJ we denote}

by LP(03A9;RN)

the Banach space of

mappings

u : Q - u ⁼

(ui , ... , u_; ) ,

^{such that uz e}

LP (Q)

^{for each}

^I,

^{with norm}

> =

~N i=1 ~ui~Lp(03A9). Similarly,

^{we denote}

by

^the

usual Sobolev space of

mappings

^{u e}

LP (Q ; R°" )

all of whose distributional

derivatives ~ui ~xj

⁼

^{Djui, I}

^I

^{N, I j}

^n,

^belong

^to

^LP(Q).

W ,P (Q, R )

^{is a} Banach space under the ^norm

where Du ⁼ and we

define,

^as

usual, f~l’~ )

^{to be the}

closure of

f~ l’~ )

^{in the}

topology

^of

Weak and weak * convergence of sequences ^are written as ^~ and

-~,

respectively.

^{If H C P e}

MNxn,

^{then we write H + P to denote}

the

set {P

⁺

^{Q : Q}

^e

H ~ , j H = ~ j x,

^{x e}

H ~

^{for an}

integer j

^{> 0. We}

define the distant function for a set K C

by

DEFINITION 2.1.

(see Morrey

^Ball

Ball,

Currie and Olver

(BCOJ). -

A continuous

function f :

^is

quasiconvex if

for

every P E E

Co ( U;

^and every open bounded subset U c

For a

given function,

^{we can} consider its

quasiconvexification

(quasiconvex relaxation):

^,

DEFINITION 2.2.

(see Dacorogna Suppose f :

^{I~ is a}

continuous

function.

^The

quasiconvexification of f

^is

defined by

and will be denoted denoted

by Q f.

PROPOSITION 2.3.

(see Dacorogna (DaJ). - Suppose f :

^{(~ is}

continuous,

then

where SZ c is a bounded domain. In

particular

^the

in fimum

ⁱⁿ

(2.1 )

^is

independent of

the choice

of

^Q.

Annales de l’Institut Henri Poincaré - linéaire

QUASICONVEX FUNCTIONS, SO(rt) ^{AND TWO} ELASTIC WELLS 765

We use the

following

^theorem

concerning

^{the existence and}

properties

of

Young

^measures from Tartar

[T].

For results in a more

general

^context

and their

proofs,

the reader is referred to Berliocchi and

Lasry [BL],

^Balder

[Bd]

^{and Ball}

[B 13 ] .

THEOREM 2.4. - Let

(z~~~ )

^{be a} bounded sequence in

L°° (SZ;

^Then

there exist a

subsequence (z~v> )

^and

a family of probability

measures on

~s, depending measurably

^{on x E}

^SZ,

^{such that}

for

every continuous

function f :

^{I~s -~ (~.}

We say that ^vx is a trivial

Young

^{measure at x E H if v~ for some}

A e

IRs, where 8 A

is the Dirac mass at A.

Suppose

^{that H C Rn. A}

family

^of

parametrized measures {vx}x~03A9

^is

called a

Young

^{measure limit of}

gradients [BFJK, KP],

^{if it is}

generated by

^a sequence of

gradients Duj

^with bounded in

W1,p(03A9,RN).

Let r > 0 and x

E Rn,

^set

B(x, r) = {y

^e

IRn :1 y - x r~

^and

meas(B(x, r))

^{= where is the area} of the n -1 dimensional

sphere.

DEFINITION 2.5.

(The

^Maximal

Function). -

^{Let u E}

Co (Rn ).

^We

define

where we set

for

every

locally

^summable

f.

LEMMA 2.7.

Vol. 14, n ° ^6-1997.

766 _{K. ZHANG}

for

^{all x E} R". Moreover

(see [20]) if

^{p >}

1,

^then

and

if

p ^>

1,

^then

for

^{all A > 0.}

LEMMA 2.8.

Then

for

every x, y ^E

H03BB

we have

LEMMA 2.9. - Let X be a metric _space, E a

subspace of X,

^{and k a}

positive

real number. Then _any

k-Lipschitz mapping from

^{E into (~ can be}

extended to a

k-Lipschitz mapping from

^{X into I~.}

For the

proof

^see

[ET,

page

298].

DEFINITION 2.10.

(see

page

234J). -

Let SZ C I~n be _open. Let B c f~p be a Borel subset. A

mapping

is said to be a

Carathéodory function if

( 1 ) for

^{almost all x E}

S2, ~ (x, . )

is continuous on

B, (2) for

^{all a E}

B, f(., a)

is measurable on SZ.

We will not introduce the more

general

notion of normal

integrals

^to

which the Measurable Selection Theorem

applies (see [ET,

_page

234]).

THEOREM 2.1 l.

(The

measurable selection theorem

(see (ET,

page

236J).

- Let B be a compact ^subset

of

^{Rp and} g ^a

Carathéodory function of

^{03A9 x B.}

Then,

there exists ^a measurable

mapping

^{ic : SZ --~ B} such that

for

^{all x E SZ:}

Annales de l’Institut Henri Poincaré - Analyse ^linéaire

767 QUASICONVEX FUNCTIONS, SO(n) ^AND TWO ELASTIC WELLS

A direct consequence of Theorem 2.11 is the

following:

PROPOSITION 2.12. - Let B c I~p be a compact subset and let u : SZ --~ I~P be an

integrable mapping.

Then there exists a measurable

mapping

ic : B such that

for

^{all x E SZ}

We conclude this section

by giving

^the definition of

quasiconvex

^hull

QK

^{of a}

compact

^{set K C}

MNxn,

^{and some}

_simple properties

^{of it. We}

use a more restricted definition than that in

[S].

DEFINITION 2.13.

(see

^{Let K C}

MN"n

be non-empty ^and compact.

The

quasiconvex

^hull

QK of

^{K is}

defined by

PROPOSITION 2.14.

and

It is easy ^{to see} that is

quasiconvex, QK

^C

f a f (0)

^and

^QK

⁼

We may ^assume that

f ~ f (0)

^is

^compact, otherwise,

^{take the}

convex function

which is the

squared

distance function to a convex set. Therefore

+ g

^is

quasiconvex.

We claim that

(fa!

^{+ C}

^convK,

hence it is

compact.

This is easy ^{to see} because

fa f

^{> 0 and = convK. We}

^have,

^for

any fixed 1 p oo,

From

[Z,

^Theorem

1.1]

^{and its}

proof,

^{we see that for a}

compact

^{zero set}

fa f (0) corresponding

^{to a}

nonnegative quasiconvex

^function and for any

1 ~

p oo,

Hence, Ki

^c

fa f (0))

^{for every}

quasiconvex

^function

f,

^thus

K1

^c

QK.

The

proof

^is

complete.

^D

Vo!.14,n° 6-1997.

768 K. ZHANG

3. THE APPROXIMATING LEMMA

In this section we establish our main

approximation

lemma which

applies

to

general compact

^{sets in}

LEMMA 3.1. -

Suppose

^{that K C is a} compact subset. Let 1 _p oc. Let P E

MNxn be

such that

that

is,

where D ⁼

(0, 1)n

^C ~n is the unit cube. Then there exist a

minimizing

sequence bounded in such that

~c~ ^{- 0 in a bounded} sequence

(gj)

^{in and}

a constant

C(n, N, p)

^{such that}

where r~~ ^{~ 0}

as j

^~ oo,

and

Proof - Let 03C6j

^E

Co (D,

^{be a}

minimizing

sequence

as j

^{~ ~,} where ~j ^{~ 0} is a

nonnegative

sequence. Let

Kp

⁼

{A 2014 ^P,

A ~

~C},

^{and set}

769 QUASICONVEX FUNCTIONS, SO(n) ^AND TWO ELASTIC WELLS

It is easy ^{to see} that

We have

as j -

^{oo. Since}

^Kp is ^compact,

^{we see that} is bounded in

Wo ’p ( D, (~ N ) ;

ⁱⁿ

^fact,

^since

we have

Extend ~~

^to be defined in IRn as a

periodic function,

and then let

for j

⁼

1, 2, ... ,

^{and xED. It} is easy ^{to see} that is bounded in and up ^{to a}

subsequence,

~c~ ^{~ 0 in Let}

B(o,11)

^{be a ball in}

^{MN n}

^{such that}

Kp

^C

B(o,11)

^and

^2-(p-1)p

^>

Extend ~c~

by

^{zero outside}

D,

^{we see that} _u~ ^E

(~ ~’T )

uj =

(u(1)j,..., u(N)j).

^{For each}

fixed j, i,

^define

Lemma 2.8 ensures that for all _{x, y} e

H~ ,

Let be a

Lipschitz

^function

extending

^outside

HI

^with

^Lipschitz

constant not

greater

^than

C(n)A (Lemma 2.9).

^Since

HI

^{is an open set,}

we have

Vol. 14, n 6-1997.

770 K. ZHANG

for all x E

H~

^and

If

H~ _ ^~,

^{set 0 for x E D. If}

HI

ⁿ

^{D ~ ~,}

^{then we may}

also assume that

Indeed, fixing y

^E

H;

ⁿ

^D,

for every x ^E

D,

where we have used Lemma 2.7 to assert

that u~2 ~ ( g ) ~ ( M * u~.2 ) ) ( g )

^A

when _{y e}

H~ .

^{Now set}

( g~ 1 ~ , ... , g~ N ~ ) .

In order to estimate

J D ^Kp)dx,

^{we start from the}

^inequality

and find a bound for

meas ( D ~ H~ ) .

Similar to the

proof

of Lemma 3.1 in

[Z],

^we

have,

from the definition of

and

Define h :

by

Annales de l’lnstitut Henri Poincaré -

QUASICONVEX FUNCTIONS, SO(n) ^AND TWO ELASTIC WELLS 771

so that we can prove that

In

fact,

^{when >}

2 ,

^{we have a sequence of a~ >}

^0,

0 and a sequence of balls

Bk

⁼

B(x, Rk)

^{such that}

which

implies

Passing

^{to the} limit A; ~ oo in

(3.6),

^{we obtain}

(3.5) (we

have chosen

~> ^A).

From Lemma

2.6,

^{we have}

Also,

from Lemma

2.6, together

^{with the}

embedding ^theorem,

^{we have}

Vol. 14, n° 6-1997.

772 _{K. ZHANG}

as j

^{-~ oe. Therefore}

Since ~~

^is

periodic,

^{we have}

Therefore,

which

implies

^that

From the above two sets of

inequalities,

^{we have}

Hence

Annales de l’Institut Henri Poincaré - linéaire

QUASICONVEX FUNCTIONS, AND TWO ELASTIC WELLS 773

Consequently,

Hence,

Also,

^from

we deduce that

thus,

Therefore

Vol. n v 6-1997.

774 _{K. ZHANG}

Given

A,

^{B e}

MNxn,

^{if we} take

Q

^E

Kp

^{such that =}

dist ( A, Kp ),

we have

and so

Therefore,

^from

(3.11)

^{we obtain}

Now, taking

^{A = = we} have

which

yields (3.3)

^after

letting j

^{-~ oo. Also}

almost

everywhere

^{in D and}

Since

distp(P,convK)

^{is a convex} function and

distp

(P, K),

^we

have,

from the definition of

quasiconvexification,

Annales de l’Institut Henri Poincaré -

QUASICONVEX FUNCTIONS, S’O(n) ^{AND TWO ELASTIC WELLS} 775

Therefore,

thus,

From the numerical

analysis point

^of

view,

^{when we} minimize the energy in a bounded subset of

W 1 ~ °°

and compare the minimum with the

minimum,

^{we have}

proved

^the

following

^result

COROLLARY 3.2. -

Suppose

^{that K C is}

compact.

^{Then there exist}

Ci(n, N, p)

^>

^0, C2 (n, N, p)

^>

^0,

^{such that}

The

only

fact in this

Corollary

^to be remarked is that the

W 1 ~ °°

bound

of g depends

^on

K),

^{while in}

(3 .15 )

^it

depends

^{on In}

fact, they

^are

equivalent

because of the

following inequalities,

Remark 3 .1. - From the construction

of gj

^{in the}

proof

^{of Lemma}

3.1,

we see that even if a

minimizing

sequence is ^not bounded in we

may find ^a

W1,~-sequence

^{such that near the zero}

_points

^of

Qdistp(., K),

it serves as an

approximate minimizing

sequence.

4. PROOFS OF THE MAIN RESULTS

Proof of

^Theorem 1 .1 . - Let

F(P) _ ,S’O (n) )

^{for P E}

It is known

(see [Z])

^that

F-1(0) _ SO(n) .

^{Let 0 a =}

F(P)

^{for some}

P,

^{i. e.}

Vol. 14, n° 6-1997.

776 _{K. ZHANG}

By

^Lemma

3.1,

^{we find and}

(gj ) satisfying (3.1)-(3.15)

^with p ⁼ 2.

We have that

and

by (3.15)

We may ^assume

that,

up ^{to a}

subsequence,

ⁱⁿ

From the measurable selection theorem

(see Proposition 2.13),

^{for each}

i ⁼

1,2,...,

there exists a measurable

mapping Pi :

^{D -~}

SO(n),

^such

that

almost

everywhere

in D. Let

Following Sverak [S],

^we consider the

integral

for

2, j

⁼

1, 2,....

From the weak

continuity property

^of

null-Lagrangians [B I,R],

^{if we let i -~} oc ,

then j

^{-~ oo,}

Now,

^since

P2(x), Pj(x)

^E

SO(n)

^almost

everywhere,

^{we have}

where

adjPi

^{+ is the}

expansion

^of

adj (Pj

⁺

nj),

^and

Annales de l’Institut Henri Poincaré - linéaire

QUASICONVEX FUNCTIONS, ,SO(n) ^AND TWO ELASTIC WELLS 777 We then have

Here we have used the facts that

if P, Q E SO(n),

^ab

a2-~- 4 .o

for real numbers and that ⁼

P + Dgi.

From

(4.3)

^we

get

where

C(n)

^{> 0 is a constant}

depending only

^{on n.}

Since a~/~ +

Vol. 14, n° 6-1997.

778 _{K. ZHANG} and so, from

(4.4)

^and

(4.1)

^we conclude that

Now we find a lower bound of

Recalling

^{the estimate}

(3.11)

^on

Dui|pdx,

^we

^have,

^{for our}

choice of

A,

and

A,

where ~j ⁼

0, limi~~

~i ⁼

0,

^and

C(n, N, p) _ C(n),

^{N = n,}

p ⁼ 2 is a

positive

^constant.

Combining ^(4.5)

^and

(4.6),

^{we see that}

Since the

integral

^on the left hand side of

(4.7)

^is

lower-semicontinuous

as

z ^~ oc

and ui

^{-- 0 in}

R"), letting

^{z ~ oo we have}

Annales de l’Institut Henri Poincaré - Analyse ^linéaire

QUASICONVEX FUNCTIONS, SO(n) AND TWO ELASTIC WELLS 779

By applying

^the measurable selection lemma

(Theorem 2.13)

^{to the function}

Q ~ ^{for Q}

^E

SO (n),

^{we can find a} measurable

mapping Qj :

^{S~ -~}

SO(n),

^{such that}

for almost every x ^E S~. Therefore the

following inequalities

hold almost

everywhere,

and also

As Uj

^{is a}

minimizing

sequence, ^we pass ^to the limit

as j

^{-~ oo and}

we obtain

Since we also have

and when

~P~

^{> we obtain}

while,

^{when we have}

we conclude that

Vol. 14, n° 6-1997.

780 K. ZHANG

which, together

^with

(4.10) yields

for ^some

C(n)

^{> 0. Therefore we} conclude that there exists

c(n)

^{> 0}

⁽ⁱⁿ

fact

c(n)

⁼

I/C(n))

^{such that}

for all P e Mnxn. D

Proof of Corollary

1 .2. - In the

proof

of Theorem 1.1 we used the fact that if

P, Q

^E

SO(n),

It was observed

by Sverak

in

[Sv] (also

^see

[Ma])

that under the condition

(1.2),

there exists an

a(H)

^>

^0,

^{such that}

We denote

by

^the

greatest eigenvalue

^{of Hand}

we see that if

P, Q

^E

SO(n), ^{(4.11 )}

^{holds for a = l. If}

^{P, Q}

^E

SO(n)H,

.

(4.11)

^{holds for a =} This is because ^{we can} write P ⁼

PIH, Q

⁼

QIH

^{for some}

P1, Qi

^E

SO(n)

and notice that H is ^a

diagonal matrix,

^{so that}

while

Hence we reach the conclusion.

Finally,

^{if P E}

SO(n)H

^and

Q

^E

SO(n),

then we write P ⁼

RH,

^{where R e}

S‘ O ( n ),

^{and since trRH ~}

^trH,

trH R e

SO(n)

^{and H is a}

diagonal

matrix with

positive ^entries,

we see that

QUASICONVEX FUNCTIONS, SO(n) ^{AND TWO} ELASTIC WELLS 781 Since

it is clear that

(4.11)

^{is satisfied}

setting

Take

and

(4.11)

holds for all

P, Q

^{E K.} We may follow the

proof

of Theorem 1.1 to prove

Corollary

^{1.2. The}

only place

^{in the}

proof

of Theorem 1.1 where

a(H)

is involved is the last

inequality (4.3).

^{Also the constants}

C(n), c(n)

in Theorem 1.1 are

replaced by C ( n , H )

^and

c ( n, H ) , respectively.

^D

Remark 4.1. - In fact Matos

[Ma] proved

that if for some

eigenvalue A/,

^of

H,

^{we have}

then the

quasiconvex

^{hull of K} remains itself. It turns out

that,

^{it is}

enough

to assume this condition in

Corollary

1.2. What ^we need is a variation of

(4.11 ).

^Let

EE

⁼

(eij)

^{be a} matrix such that eij ^{= 0 if i}

~ j,

^{eii = E for}

some E > 0

sufficiently

small if i

~

^{k and ekk = 1. We} consider the form

Then this form is still a null

Lagrangian.

^{We also}

have,

^{for P E}

SO(n)H, Q

^E

SO(n),

when E > 0 is small

enough.

Therefore the conclusion of

Corollary

^{1.2 is}

still true under this weaker condition.

Proof of Corollary

1.3. - Let

F(P)

⁼

Qdist2(P, SO(n)),

^{and let}

P E Then

by proposition

^{2.15 we have}

Vol. 14, n° 6-1997.

782 K. ZHANG

and since

using

^{Theorem 1.1 we} obtain

The

proof

^is

^finished

^for

SO(n).

^{The other case} is similar. 0

Proof of

Theorem 1.4. - We

only give

^a

proof

^for

f :

The

proof for g

is similar. is a

family

^of

Young

^{measure limit of}

gradients supported on K j.

^{If we can show that} v~ is trivial for almost every

x, the

proof

^{will be}

^finished.

It is known

[Z, KP]

^{that if a}

Young

^measure

limit of

gradients

^has

bounded support,

^{then the}

Young

measure can be

generated by

^a bounded sequence in

W 1 ~ °° .

Next we notice that if E is small

enough,

^{then the}

mapping

^{Y = X +}

^f(X)

^from

^SO(n)

^to

^Kj

^is

^invertible

and the inverse is continuous. Let

Q(.) :

^: SZ --~ Mn x n be measurable. If

we

apply

the measurable selection

theorem,

^we may find a measurable

mapping

P : SZ --~

Kj,

^{such that}

/P(x) -

^T

dist(Q(x), Kj),

^where

P(x)

⁼

R(x)

⁺

f(R(x)),

^{and R : SZ -~}

_SO(n)

^is measurable.

Now,

^let be a bounded sequence in such that the

Young

^measures

generated by (Duj)

^is

supported

ⁱⁿ

K f

^and Uj

-~ ~c

in

W 1 ~ °° ( S~, (~=’~ ) .

^For

each

j,

^we may find ^a measurable

mapping Rj : SO(n)

^{such that}

almost

everywhere. Let nj

⁼

Rj(x) - f(Rj(x)).

^{As in the}

proof

of Theorem

1.1, setting

we have

limi~~ limi~~ Iij

⁼ 0. We also have

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

QUASICONVEX FUNCTIONS, SO(n) ^{AND TWO ELASTIC WELLS} 783

where

C(Rj, f(Rj), nj)

is defined

by

^the

expansion

of the determinants

Therefore we

have,

^{due to} the fact that

Rj

^E

~S’O(n)

^and

f

^satisfies

(1.5),

that

Since

Duj = Rj

⁺

f(Rj)

⁺ nj, ^from

(1.5),

^{and if we choose E}

1/2,

we have

Thus we

have,

Vol. 14, n° 6-1997.

784 K. ZHANG

Obviously,

Choosing

^{E =}

min~l/2,1/(4~’(n))~,

^{we see that if we combine}

^(4.13)

and

(4.14),

^{we have}

Since

I

^-~

^0, when j --~

^{0 in} for any p ^>

1,

^and

(Duj)

^is

bounded in

L°°, passing

^to the limit in i ^--~ _oo, and

using

the fact that the functional on the left hand side of

(4.15)

^{is lower}

semicontinuous,

^then

letting j

^{-~ oc in}

(4.15),

^{we see that}

Therefore vz ⁼

bD.t~ ~ ~~

^almost

everywhere.

The

proof

^for

K~

^{is similar. D}

The

proof

^for

Corollary

1.5 is similar to that for Theorem 1.1 and it is left to the reader.

ACKNOWLEDGEMENTS

This work is

supported by

the Australian

government through

^the

Australian Research Council. The author would thank the referee for many

helpful suggestions.

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(Manuscript received October 2, 1995;

revised February 5, 1996. )

Vol. 14, n" 6-1997.